Climbing Stairs
You are climbing a staircase with n steps. Each time you can climb 1 or 2 steps. How many distinct ways can you reach the top?
Example:
3
3
- To reach the top of a staircase with 3 steps, we can climb 1 or 2 steps at a time, so we consider all possible combinations of 1 and 2 steps.
- The distinct ways to reach the top are:
- 1 step + 1 step + 1 step
- 1 step + 2 steps
- 2 steps + 1 step
- We count the number of distinct ways, which is 3 in this case.
- The final output is the total count of distinct ways, which is 3​.
Constraints:
- 1 <= n <= 45
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Background Knowledge
The "Climbing Stairs" problem is a classic example of a dynamic programming problem. Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems, solving each subproblem only once, and storing the solutions to subproblems to avoid redundant computation. This approach is particularly useful for problems that have overlapping subproblems or optimal substructure. In the context of the "Climbing Stairs" problem, we can break down the problem into smaller subproblems by considering the number of ways to reach each step.
The key concept in dynamic programming is the idea of memoization, which involves storing the solutions to subproblems in a memory-based data structure (such as an array or hash table) to avoid recomputing them. This approach can significantly reduce the time complexity of the solution. Another important concept is the idea of recurrence relations, which define the relationship between the solution to a problem and the solutions to its subproblems. In the case of the "Climbing Stairs" problem, we can define a recurrence relation to represent the number of ways to reach each step based on the number of ways to reach the previous steps.
To solve dynamic programming problems, it's essential to identify the state and transition of the problem. The state represents the current situation or status of the problem, while the transition represents the action or decision that leads to the next state. In the "Climbing Stairs" problem, the state can be represented by the current step, and the transition can be represented by the decision to climb 1 or 2 steps.
Algorithm/Approach
The general approach to solving the "Climbing Stairs" problem involves using dynamic programming to build up a solution by considering the number of ways to reach each step. The algorithm pattern involves:
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